Several methods are known for determining conditional probabilities. For example, in the multipoint statistical method, conditional probabilities are evaluated by enumerating events for given configurations of points in a training image. That method is described in particular in application WO 2006/023597.
The method of simulation by multinominal random fields as developed by P. Bogaert (see for example “Spatial prediction of categorical variables, the Bayesian maximum entropy approach”, published in the journal: Stochastic environmental research and risk assessment, Vol. 16, No. 6, December 2002, pp. 425-448, Springer Berlin/Heidelberg, DOI: 10.1007/s00477-002-0114-4), makes use of a step of calculating joint probabilities. A joint probability is a probability of obtaining a given event, the event consisting in a configuration state, i.e. an arrangement of states for a categorical variable located in space.
Thus, in the context of simulating a geological zone, a joint probability is the probability of simultaneously observing N states of a categorical variable ck0, cki, . . . , ck(N-1) at N respective nodes x0, x1, . . . , xN-1 of the grid. If consideration is given to a pair of nodes, then this is referred to as bivariate probability rather than joint probability.
On the basis of such joint probabilities, it is possible to determine conditional probabilities by using Bayes' theorem.
In order to establish joint probability tables, use is conventionally made of the method of maximizing likelihood. In the algorithm proposed by Bogaert, likelihood is maximized by using the iterative proportional fitting (IPF) algorithm as described for example by W. E. Deming and F. F. Stephan in “On a least square adjustment of a sample frequency table when the expected marginal totals are known”, Annals of Mathematical Statistics, Vol. 11, p. 427, 1940, but by applying it to the geostatistical context. That method makes use of bivariate probabilities for simultaneously observing two states of a categorical variable ck, ck′ at two respective nodes xi, xi′ of the grid.
It is thus necessary to calculate bivariate probabilities for each pair ck, ck′ of states of the categorical variable and for each pair of nodes xi, xi′ of the nodes in the neighborhood under consideration. This step is relatively expensive in calculation time and in memory, and in practice it becomes very difficult to implement, in particular when using more than twenty neighboring nodes and more than three categorical variable states.